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Seifert Matrix


Given a Seifert form f(x,y), choose a basis e_1, ..., e_(2g) for H_1(M^^) as a Z-module so every element is uniquely expressible as

 n_1e_1+...+n_(2g)e_(2g)
(1)

with n_i integer. Then define the Seifert matrix V as the 2g×2g integer matrix with entries

 v_(ij)=lk(e_i,e_j^+).
(2)

For example, the right-hand trefoil knot has Seifert matrix

 V=[-1 1; 0 -1].
(3)

A Seifert matrix is not a knot invariant, but it can be used to distinguish between different Seifert surfaces for a given knot.


See also

Alexander Matrix

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References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 200-203, 1976.

Referenced on Wolfram|Alpha

Seifert Matrix

Cite this as:

Weisstein, Eric W. "Seifert Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SeifertMatrix.html

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